The chief advantage of GKP states is that qubit Clifford operations map to CV Gaussian operations, which can be implemented deterministically and easily using linear optical elements, homodyne detection, and Gaussian states of light [1]}.
In sec:opticalelements, we provide optical circuits for the application of GKP qubit gates in detail.
Briefly, Pauli \(X\)  and \(Z\)  gates on GKP qubits correspond to displacements along \(q\)  and \(p\)  by \(\sqrt{\pi }\) , respectively.
The Hadamard gate is simply a \(\pi /2\)  rotation in phase space, implementable by a phase-shifter.
The qubit phase gate \(\sqrt{Z}\)  corresponds to phase-space shear, which can be implemented by a single-mode squeezer sandwiched between two phase-shifters.
The qubit \(CNOT\)  and \(CZ\)  gates correspond to \(CX\)  and \(CZ\)  gates in the CV domain, respectively, which in turn can be broken down into a pair of single-mode squeezers between two beam-splitters.
Deterministic all-optical entangling gates are a distinct advantage of the GKP encoding over dual-rail encoding schemes [2]}.
Moreover, qubit Pauli measurements correspond simply to homodyne measurements, which operate at faster speeds and higher efficiencies than photon-counting detectors [3]}, [4]}.
Finally, non-Clifford operations require a non-Gaussian resource.
Unitary implementations of the \(T\)  gate can be achieved using a cubic-phase interaction, though this is difficult in practice [5]}, and does not perform well for finite-energy states [6]}.
As an alternative, \(T\)  gates can be performed through gate teleportation by preparation of a GKP magic state [1]}, which we also review in sec:opticalelements.
